Optical Spectral Weight, Phase Stiffness and Tc Bounds for Trivial and Topological Flat Band Superconductors

TL;DR

This paper derives exact bounds on optical spectral weight and superfluid stiffness in flat band superconductors, revealing how interactions influence superconductivity in trivial and topological flat band systems.

Contribution

It provides the first exact bounds on optical spectral weight and superfluid stiffness for flat band superconductors, including trivial and topological cases, linking these to interactions and Wannier function properties.

Findings

Bounds for optical spectral weight in trivial flat bands.

Upper bounds for superfluid stiffness and Tc in flat band superconductors.

Comparison of bounds with mean-field and quantum Monte Carlo results.

Abstract

We present exact results that give insight into how interactions lead to transport and superconductivity in a flat band where the electrons have no kinetic energy. We obtain bounds for the optical spectral weight for flat band superconductors, that lead to upper bounds for the superfluid stiffness and the 2D $T_c$. We focus on on-site attraction $|U|$ on the Lieb lattice with trivial flat bands and on the $\pi$-flux model with topological flat bands. For trivial flat bands, the low-energy optical spectral weight $\widetilde{D}_\text{low} \leq \widetilde{n} |U| \Omega/2$ with $\widetilde{n} = \min\left(n,2-n\right)$, where $n$ is the flat band density and $\Omega$ the Marzari-Vanderbilt spread of the Wannier functions (WFs). We also obtain a lower bound involving the quantum metric. For topological flat bands, with an obstruction to localized WFs respecting all symmetries, we again obtain an upper bound for $D_{\rm low}$ linear in $|U|$. We discuss the insights obtained from our bounds by comparing them with mean-field and quantum Monte-Carlo results.